Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

learn more… | top users | synonyms (1)

0
votes
1answer
52 views

Accurately calculating large factorials

I need to calculate the exact value of $1,000,000,000!$ It can't be an approximation. As far as I am aware, the only way to do this would be to calculate it starting from the beginning $(1\cdot2\cdot3\...
1
vote
1answer
40 views

Do negative binomials imply negative factorials exist?

I've seen the following identity: $$\binom{-n}{k} = (-1)^k\binom{n+k-1}{k}$$ So I tried to derive it, assuming negative factorial was a real concept, having it extend down to negative infinity: $$\...
0
votes
1answer
14 views

Prove that: ${^{n}\mathrm{C}_{k}} = {^{n-1}\mathrm{C}_{k-1}}+{^{n-1}\mathrm{C}_{k}}$ [duplicate]

Question asks to prove: ${^{n}\mathrm{C}_{k}} = {^{n-1}\mathrm{C}_{k-1}}+{^{n-1}\mathrm{C}_{k}}$ My Steps: $$\begin{align*}\frac{(n-1)!}{(n-k-2)!(k-1)!} + \frac{(n-1)!}{(n-k-1)!(k)!} & = \...
1
vote
1answer
33 views

Simplifying $\frac{^n\mathrm{C}_k}{^n\mathrm{C}_{k-1}}$

Question asks to simplify: $$\frac{^n\mathrm{C}_k}{^n\mathrm{C}_{k-1}}$$ I have a few steps but not sure if its correct. $$\begin{align*}\frac{(n)!}{(n-k)!(k)!} \bigg/ \frac{(n)!}{(n-k-1)!(k-1)!}...
-3
votes
2answers
59 views

simplify factorials: $\frac{(k-1)!}{(k+2)!}$ [duplicate]

Question: simplify $$\frac{(k-1)!}{(k+2)!}$$ What I did was: $$\frac{(k - 1)!k!}{(k + 2)! \cdot (k + 1)!}$$ This I did following the rule $n! = n \times (n - 1)!$. can this be simplified ...
0
votes
2answers
63 views

Simplifying factorials: $\frac{(n-1)!}{(n-2)!}$

Question: simplify $$\frac{(n-1)!}{(n-2)!}$$ What I did was: $$\frac{(n - 1)!}{(n - 2)! \times (n - 3)!}$$ This I did following the rule $n! = n \times (n - 1)!$. But my answer just doesn't look ...
3
votes
1answer
50 views

Proof of Stirling's Formula using Trapezoid rule and Wallis Product

I need a proof of stirling's formula which uses the riemann's sum and trapezoid approximation to come up with $ \frac {n!}{(n/e)^n \sqrt n}$ $ \rightarrow C$ where $C$ is derived from Wallis product. ...
4
votes
1answer
25 views

Find an explicit map with certain combinatorial properties

The following combinatorial problem popped up in a totally uncombinatorial context: Let $\mathcal{P}$ denote the power set of a set and let $k \in \mathbb{N}$. Is there a map $c: \mathcal{P}(\{1,2,\...
1
vote
1answer
17 views

Find a map on a power set with certain combinatorial properties

The following combinatorial problem popped up in a totally uncombinatorial context: Let $\mathcal{P}$ denote the power set of a set and let $k \in \mathbb{N}$. Is there a map $c: \mathcal{P}(\{1,2,\...
1
vote
2answers
26 views

Analytic continuation and how it relates to the gamma function.

I am familiar with factorials, and I have read about the gamma function. From what I understand, the gamma function extends the concept of the factorial to complex numbers by nature of being an ...
3
votes
2answers
80 views

Finite summation including binomial coefficients and double factorials

I came across the following summation: $$ \sum_{k=0}^n\frac{(-1)^k(2k)!!}{(2k+1)!!}\dbinom{n}{k}\,\,\,\,(n\in\mathbb{N}). $$ $\tbinom{n}{k}$ are binomial coefficients, $n!/k!(n-k)!$. Mathematica told ...
0
votes
1answer
19 views

recurrence relation - How to determine pattern for an even or odd or different type of factorial

Hi I am having trouble on how to solve for the odd terms of recurrence relation in terms of exponential and factorials. How are you able to see a pattern to simplify a non standard factorial. This ...
2
votes
1answer
56 views

Is there a more concise expression of this product?

In a longer computation, I have stumbled upon the following product, where $k,r \in \mathbb{N}_0$ are fixed numbers: $$\prod_{0 < i_0<i_1<\dots<i_r\leq k} (i_r-i_{r-1})(i_{r-1}-i_{r-2})\...
2
votes
1answer
40 views

notation for factoraling a factorial? (since one cannot do n!!)

I was thinking about how to get a number to be larger than graham's number very easily... came up with "factoraling" a factorial. However the notation n!! means something completely different. And I ...
6
votes
4answers
166 views

Proof of the summation $n!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$?

I was going through a Number Theory book the other day and found this question. It asked for the proof of the following equation: $$n!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$$ I tried hard but ...
2
votes
2answers
73 views

Number of zeros at the end of $10^{2}!+ 11^{2}!+12^{2}! \cdots+99^{2}!$

How do I find the number of zeros at the end of the Integer $$10^{2}!+ 11^{2}!+12^{2}! \cdots+99^{2}!$$ Answer provided for this question is $24$
1
vote
0answers
23 views

When is $\frac{2 n f(n)}{n !}$ in the order of some fixed power of $n$?

I would like to know when $\frac{2 n f(n)}{n !}$ is $O (n^b)$ where $b$ is a constant. Here, $n$ is a positive integer. My attempt: $$ \frac{2 n f(n)}{n !} = \frac{2 n f(n)}{\sqrt{2 \pi n} (\frac{n}{...
0
votes
2answers
77 views

Factorial sum estime

Prove that: $$\displaystyle \sum_{n=m+1}^\infty \dfrac{1}{n!} \le \dfrac{1}{m\cdot m!}$$ I have tried induction on $m$ but it does not work very well. Any suggestion?
5
votes
5answers
1k views

Approximation of log(n!)

I just finished calculus 1 (derivative and integral) then I take another course on calculus 2. In the video the professor talks about the the series $$\frac{n!}{(\frac{n}{e})^n}$$ He shows the ...
2
votes
1answer
129 views

A couple of series questions that I just can't figure out (Calc 2)

Show that $$ \begin{align} \left(\frac{\pi}{2}\right)^2\left[\int_0^{\pi/2}\cos^{2n}t\ dt-\int_0^{\pi/2}\cos^{2n+2}t\ dt\right]&=\frac{\pi^3}{8}\left[\frac{(2n-1)!!}{(2n)!!}-\frac{(2n+1)!!}{(2n+...
1
vote
1answer
53 views

Find all nonnegative integers $m$ and $n$ such that $m!+1=n^2$. [duplicate]

This question is inspired by Subgroup of Order $n^2-1$ in Symmetric Group $S_n$ when $n=5, 11, 71$. Find all nonnegative integers $m$ and $n$ such that $m!+1=n^2$. We know that $(m,n)=(4,5)$, $(...
5
votes
3answers
88 views

Factorial Proof by Induction Question? [duplicate]

$\text{Use the PMI to prove the following for all natural numbers n.}$ $ \frac{1}{2!} + \frac{2}{3!} + \cdot \cdot \cdot + \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!} $ So for this question I get ...
1
vote
1answer
25 views

Finding the remainder of an exhausted $m!$

For any prime $p$, let $n_p(m)$ denote the exponent of $p$ in the factorisation of $m!$, i.e. $m!=p^{n_p(m)}\cdot k$ with $p\not\mid k$. Is wonder if there is a general formula of $\frac{m!}{p^{n_p(...
1
vote
1answer
31 views

From $ \frac{\left(n\cdot \:n!+1\right)\left(n+1\right)}{\left(n+1\right)\left(n+1\right)!+1} $ to $ \frac{n+\frac{1}{n!}}{n+1+\frac{1}{n!}} $?

Good evening to everyone. I have an expression that I don't know how to arrive at. $$ \frac{\left(n\cdot \:n!+1\right)\left(n+1\right)}{\left(n+1\right)\left(n+1\right)!+1} $$ to $$ \frac{n+\frac{1}{n!...
0
votes
0answers
78 views

Closed-from for the series: $\sum_{k=0}^{\infty} \frac{1}{(k!)!}$

As the title says, I'm wondering whether there is any known closed-from for the following series: $$\sum_{k=0}^{\infty} \frac{1}{(k!)!}$$ Here I don't mean the double factorial (treated here) ...
1
vote
1answer
61 views

Is there a closed-form for $\sum_{k=0}^{\infty} \frac{1}{(k!)^2}$?

As the title says, I'm wondering whether there is any known closed-from for the following series: $$\sum_{k=0}^{\infty} \frac{1}{(k!)^2}$$ Trying on WolframAlpha, I get the value $2....
0
votes
1answer
23 views

How to simplify modular exponential expressions with factorial as exponents?

Said I have the following expression: n = 39^50! mod 2251 By Fermat's Little Theorem: 39^2250 = 1 mod 2251 Solving: 2250 = 50.45 n = 39^50.49.48.47.46.45.44! mod 2251 Let b = 49.48.47.46.44! , ...
5
votes
2answers
168 views

Can there be only one extension to the factorial?

Usually, when someone says something like $\left(\frac12\right)!$, they are probably referring to the Gamma function, which extends the factorial to any value of $x$. The usual definition of the ...
78
votes
10answers
8k views

Is $0! = 1$ because there is only one way to do nothing?

The proof for $0!=1$ was already asked at here. My question, yet, is a bit apart from the original question. I'm asking whether actually $0!=1$ is true because there is only one way to do nothing or ...
1
vote
1answer
101 views

How to prove $\frac{(2n)!(2m)!}{n!m!(n+m)!}$ is an integer by strictly using my method?

I have to prove that $$\frac{(2n)!(2m)!}{n!m!(n+m)!}$$ is always an integer. I already have seen the same question here-Prove that for all non-negative integers $m,n$, $\frac{(2m)!(2n)!}{m!n!(m + n)!}...
0
votes
2answers
91 views

Prove that $\log(n!)\leq(\log(n))!$

Prove that $\log(n!)\leq(\log(n))!$ My attempt: I read somewhere that $n\leq\log(n!)\leq(\log(n))!$. But when I used calculator $\log(n!)$ can not be less than or equal to $(\log(n))!$. ...
1
vote
0answers
24 views

Function order of Logarithms

How can I use Stirling's approximation to trying to find the function order of $ceil(log(logx))!$ ? My main goal is to finding it's order of complexity but my main issue is that I'm not sure on how to ...
1
vote
2answers
68 views

Find the divisors of $5040$ in the Plato's dialogue “Theaetetus”

In the Plato's dialogue "Theaetetus", at a certain point, we have the following "problem" \begin{align*} 5040 &= 7! \\ &= 1\times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \\ &= 2 \...
5
votes
2answers
166 views
+50

Find last 5 significant digits of 2017!

Since there are less powers of $5$ than of $2$ and since $10 = 2 \cdot 5$, I counted the number of zeros in $2017!$: $\left \lfloor{ \frac{2017}{5^1}}\right \rfloor + \left \lfloor{ \frac{2017}{5^2}}\...
1
vote
1answer
73 views

Conjecture concerning modular arithmetic

Below $0\notin\mathbb N$. I want a proof or a counter-example of the following (corrected) conjecture: Suppose $p$ is the smallest prime dividing $n\in\mathbb N$ and suppose $kn+ap=m!$, where $...
8
votes
1answer
442 views

Conjecture about primes and the factorial: for all primes $p>5$, must there exist a prime $q<p$ such that $q\equiv m!\pmod p$ for some $2<m<p$?

Below $0\notin\mathbb N$. Further corrected conjecture: For all prime numbers $p>5$ there exist a prime number $q<p$ such that $q\equiv m!\!\pmod p$, $2<m<p$. or Given a prime ...
0
votes
1answer
29 views

Perfect square from a multiple of factorials

I have a problem with this question John writes the number 1!, 2!, 3!, ... , 199!, 200! on a whiteboard. John then erases one of the numbers. John then multiplied the remaining 199 numbers. He found ...
7
votes
4answers
179 views

Why does $(128)!$ equal the product of these binomial coefficients $128! = \binom{128}{64}\binom{64}{32}^2 \dots \binom21^{64}$?

I'm working through some combinatorics practice sets and found the following problem that I can't make heads or tails of. It asks to prove the following: $$128! = \binom{128}{64}\binom{64}{32}^2\...
1
vote
2answers
66 views

Restrictions on Factorial Usage

I had always understood that the factorial n! was defined as $$\prod_{k=1}^nk$$ However, this leaves several questions: Why does 0! exist?* By extension, why can't you take the factorial of a ...
6
votes
3answers
229 views

Seeking closed form for infinite sum $\sum \limits_{ n=1 }^{ \infty }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! } }$

$\displaystyle \sum _{ n=1 }^{ \infty }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! } }$ is approximately $.5229461921333351$ but I've been assured that there is a closed form for this ...
0
votes
0answers
20 views

Can Euler's generalization of factorial be done for double factorial?

Can Euler's generalization of factorial be done for double factorial? Euler's generalization of factorial to non-integer values is $t! =\lim_{n \to \infty} \dfrac{n!n^t}{\prod_{k=1}^n(t+k)} $. I ...
1
vote
2answers
57 views

How to find the Summation of series of Factorials?

$$1\cdot1!+2\cdot2!+\cdots+x\cdot x! = (x+1)!−1$$ I don't understand what's happening here. The given sum of factorials is generalized into a single term. Could somebody please help me finding the ...
12
votes
1answer
117 views

The divergent sum of alternating factorials

So I came across this exposition of a paper by Euler here where Euler is trying to sum the divergent sum: $$s = 1 - 1 + 2! - 3! + 4! \dots = \sum_{k\geq 0}(-1)^k k!.$$ There are a couple of questions ...
0
votes
0answers
24 views

Tight lower bound on falling factorial

I have the term $$p=\left(\frac{1}{n-b}\right)^a\cdot[n]_a, 0<b<a,\ b \in \mathbb{R} \text{ fixed}$$ and want to find a tight lower bound such that I will then be able to solve for n. For ...
1
vote
2answers
52 views

Expanding a factorial

Can you explain me how we got this identity? $$\frac{1}{(3n)!}$$ the same as $$\frac{(3n)!}{(3n+3)!}$$ I have been trying to expand, but didn't get the same. Thanks.
1
vote
1answer
76 views

Maybe this inequality holds? $x!-y!>x^n$?

Let $x,y,n$ be postive integers such that $x\ge 2y,y>n,n>3$ I conjectured that $$\color{red}{x!-y!\ge x^n}$$ Now, I claim that $$\color{red}{x!-y!=y![x(x-1)(x-2)\cdots(y+1)-1]\ge (n+1)![x(x-1)(...
14
votes
4answers
1k views

Evaluate $\frac{0!}{4!}+\frac{1!}{5!}+\frac{2!}{6!}+\frac{3!}{7!}+\frac{4!}{8!}+\cdots$

$$\frac{0!}{4!}+\frac{1!}{5!}+\frac{2!}{6!}+\frac{3!}{7!}+\frac{4!}{8!}+\frac{5!}{9!}+\frac{6!}{10!}+\cdots$$ This goes up to infinity. Trying finite cases may help. My Attempt:It seems that it is ...
1
vote
1answer
48 views

Is my proof to proove that $\frac{n!}{p_1!p_2!p_3!…p_m!}\in \mathbb{N}$ valid?

I wish to prove that $\frac{n!}{p_1!p_2!p_3!....p_m!}$ is an integer, were, $p_1+p_2+p_3+...+p_m=n$ and $p_i, n\in \mathbb{N}$. Pleace do check the validity of my proof. Let us consider the following ...
0
votes
3answers
82 views

Limit of $\binom{n}{k}/n^k$

How to prove that, $n$ approach infinity and $0\leq k\leq n$ : $$ \lim_{n\to\infty} \dfrac{1}{k!}\times\dfrac{n!}{(n-k)!\times n^k} = \dfrac{1}{k!}$$ I have no idea to begin the proof... Thanks in ...
3
votes
1answer
63 views

Is $S_i(1,2,\dots,p-2) \equiv 1 \pmod{p}$ for all values of $i$ whenever $p$ is prime?

Let $S_i(x_1,x_2,\dots,x_n)$ denote the $i$th elementary symmetric polynomial in $n$ variables. Is $S_i(1,2,\dots,p-2) \equiv 1 \pmod{p}$ for all values of $i$ from $0$ to $(p-2)$ whenever $p$ is ...